Determine if the function below is continuous.

A. not continuous, 1 hole
B. continuous
C. not continuous, > 2 holes
D. not continuous, 2 holes

Answer:

y = 8√2

Step-by-step explanation:

First, we can use the 30-60-90 triangle rule on the rightmost triangle. This states that:

So, we can solve for the middle line (I will call it L) on this triangle:

L = 16 / 2

L = 8

Next, we can solve for y using the rule with 45-45-90 triangles:

So, we can solve for y:

y = 8 · √2

y = 8√2

The president of Doerman Distributors, Inc., believes that 27% of the firm's orders come from first-time cu:

sample of 100 orders will be used to estimate the proportion of first-time customers. Use z-table.

0.30. What is the sampling distribution of p for this study?

a. Assume that the president is correct and p

=

Hint(s) Che

A normal distribution because np and n(1-p) are both greater than 5 v

b. What is the probability that the sample proportion will be between 0.20 and 0.40 (to 4 decimals)?

c. What is the probability that the sample proportion will be between 0.25 and 0.35 (to 4 decimals)?

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Partially Correct

Answer:

Step-by-step explanation:

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a. Since the sample size is large enough (n=100) and the expected number of first-time customers (np=27) and the expected number of repeating customers (n(1-p)=73) are both greater than 5, we can assume that the sampling distribution of the sample proportion p is approximately normal with mean μ=p=0.27 and standard deviation σ=sqrt[p(1-p)/n]=sqrt[(0.27)(0.73)/100]=0.046.

b. To find the probability that the sample proportion will be between 0.20 and 0.40, we need to standardize the values using the formula z = (x - μ) / σ, where x is the sample proportion, μ is the population proportion, and σ is the standard deviation of the sampling distribution of p. Then we can use a standard normal distribution table (z-table) to find the corresponding probabilities.

z(0.20) = (0.20 - 0.27) / 0.046 = -1.52

z(0.40) = (0.40 - 0.27) / 0.046 = 2.83

Using the z-table, the probability of getting a z-score between -1.52 and 2.83 is approximately 0.9747. Therefore, the probability that the sample proportion will be between 0.20 and 0.40 is 0.9747 (to 4 decimals).

c. To find the probability that the sample proportion will be between 0.25 and 0.35, we can follow the same steps as in part b:

z(0.25) = (0.25 - 0.27) / 0.046 = -0.43

z(0.35) = (0.35 - 0.27) / 0.046 = 1.74

Using the z-table, the probability of getting a z-score between -0.43 and 1.74 is approximately 0.5267. Therefore, the probability that the sample proportion will be between 0.25 and 0.35 is 0.5267 (to 4 decimals).

Answer:

The quotient of the rational expressions shown below is:

(x²-4-4x+4) / 1

Simplifying the numerator:

(x²-4-4x+4) = (x²-4x) + (4-4) = x(x-4)

Therefore, the quotient is:

x(x-4) / 1 = x(x-4)

Step-by-step explanation:

To solve for a in terms of other variables, a = u + k - b

Subject of formula is a topic in mathematics that involves expressing a required variable in terms of other variables in a given equation. This requires the application some mathematical principles so as to get the final expression.

From the given question, we have;

u = a - k + b

to solve for a, add k and -b to the two sides of the equation.

Thus we have;

u + k -b = a - k + b + k - b

u + k - b = a

Therefore,

a = u + k - b

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The solution is w = (3,1,-4). The solution is q = (3, 9, -4). The coordinates of vector B in the basis B = (-8, 1, 0, 8).

To find w, we need to express [w]s in terms of the standard basis. We can do this by finding the change of basis matrix from S to the standard basis, and then multiplying it by [w]s. The change of basis matrix from S to the standard basis is given by

[3 2 1] [1 0 0]

[1 5 4] = [0 1 0]

[-4 6 8] [0 0 1]

Multiplying this matrix by [w]s = [8 5 -8]ᵀ, we get

[1 0 0] [8] [3]

[0 1 0] x [5] = [1]

[0 0 1] [-8] [-4]

Therefore, [w] = (3,1,-4).

To find q, we need to express [q]s in terms of the basis {x² + 1, x² - 1, 2x - 1}. We can do this by solving the system of equations

q = a(x² + 1) + b(x² - 1) + c(2x - 1)

Substituting x = 1, we get

5 = 2a - 2b + c

Substituting x = -1, we get

5 = 2a + 2b - c

Substituting x = 0, we get:

5 = b - c

Solving these equations, we get a = 3, b = 9, and c = -4. Therefore, [q] = (3, 9, -4).

To find B, we need to express [B]s in terms of the basis {[3 6], [0 -1], [0 -8], [1 0]}. We can do this by finding the change of basis matrix from S to this basis, and then multiplying it by [B]s. The change of basis matrix from S to this basis is given by

[3 0 0 1] [1 0 0 0]

[6 -1 -8 0] = [0 1 0 0]

[0 0 0 0] [0 0 0 1]

[0 0 0 0] [0 0 1 0]

Multiplying this matrix by [B]s = [-8 1]ᵀ, we get

[1 0 0 0] [-8] [-8]

[0 1 0 0] x [1] = [1]

[0 0 0 1] [0] [0]

[0 0 1 0] [-8] [8]

Therefore, [B] = (-8, 1, 0, 8).

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Step-by-step explanation:

Let's find out:

ax - bx + y = z        subtract 'y' from both sides of the equation

ax-bx = z-y             reduce L side

x ( a-b)  = z-y           divide both sides by (a-b)

x = (z-y) / (a-b)         Done.

Two norms on a linear space X are equivalent if they define the same topology, meaning they have the same open sets. To show ||x||p and ||x||[infinity] are equivalent in Rⁿ, use the inequalities max |xi| ≤ (Σ [tex]|xi|p)^{1/p}[/tex] ≤ [tex]n^{1/p}[/tex] max |xi|.

To show that two norms on a linear space X are equivalent if and only if they have the same open sets,

If two norms on X are equivalent, then they have the same open sets.

Let ||·|| and ||·||' be two equivalent norms on X, meaning that there exist positive constants c1 and c2 such that for any x in X,

c1||x|| <= ||x||' <= c2||x||

To show that ||·|| and ||·||' have the same open sets, we need to prove that a set U is open with respect to ||·|| if and only if it is open with respect to ||·||'.

Suppose U is open with respect to ||·||. Let x be any point in U, and let r be a positive real number such that the open ball B(x, r) = {y in X : ||y - x|| < r} is contained in U. We want to show that there exists a positive real number r' such that the open ball B'(x, r') = {y in X : ||y - x||' < r'} is also contained in U.

Let c = c2/c1, and choose r' = r/c2. Then, for any y in B'(x, r'), we have

||y - x||' <= c2||y - x||/c2 = ||y - x||

Therefore, y is also in B(x, r), which implies that y is in U. Hence, U is open with respect to ||·||'.

Conversely, suppose U is open with respect to ||·||'. Let x be any point in U, and let r' be a positive real number such that the open ball B'(x, r') = {y in X : ||y - x||' < r'} is contained in U. We want to show that there exists a positive real number r such that the open ball B(x, r) = {y in X : ||y - x|| < r} is also contained in U.

Let c = c2/c1, and choose r = c1r'. Then, for any y in B(x, r), we have

||y - x||' <= c2||y - x||/c1 <= c2r/c1 = r'

Therefore, y is also in B'(x, r'), which implies that y is in U. Hence, U is open with respect to ||·||.

In Rⁿ, show that the following norms are equivalent

||x||p = (Σ[tex]|xi|^p)^{1/p}[/tex] and ||x||[infinity]: = max |xi|

i=1 1≤i≤n

To show that the two norms are equivalent, we need to show that there exist positive constants c1 and c2 such that c1||x||[infinity] ≤ ||x||p ≤ c2||x||[infinity] for all x in Rⁿ.

First, we will show that c1||x||[infinity] ≤ ||x||p. Let x be any element in Rⁿ. Then,

||x||[infinity] = max{|x1|, |x2|, ..., |xn|} ≤[tex]|x1|^p + |x2|^p + ... + |xn|^p[/tex] = Σ[tex]|xi|^p[/tex]

Since p > 0, we can take the p-th root of both sides to get

||x||[infinity] ≤ (Σ[tex]|xi|^ \infty)^{1/p}[/tex] = ||x||p

Therefore, c1 = 1 is a valid constant.

Next, we will show that ||x||p ≤ c2||x||[infinity]. Let x be any element in R^n. Then,

||x||p = (Σ[tex]|xi|^p)^{1/p}[/tex] ≤ (Σ[tex]|xi|^ \infty)^{1/p}[/tex]=[tex]n^{1/p}[/tex] ||x||[infinity]

Therefore, c2 = [tex]n^{1/p}[/tex] is a valid constant.

Since we have found positive constants c1 and c2 such that c1||x||[infinity] ≤ ||x||p ≤ c2||x||[infinity], we have shown that the two norms are equivalent.

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Developing a well-organized work schedule is crucial for the success of the facility and ensures that the business operations run smoothly, resulting in increased customer satisfaction, employee engagement, and profitability.

Developing the work schedule for the associates is the 2nd most impactful scenario as it directly affects the productivity and efficiency of the facility. A well-planned and balanced work schedule ensures that the right number of associates are scheduled at the right time to meet the demands of the business while minimizing unnecessary labor costs. Failure to create an effective schedule can result in understaffing, overstaffing, or scheduling conflicts, which can lead to a decrease in productivity and customer satisfaction.

The work schedule also affects employee satisfaction, as they need to balance their personal lives with their work schedules. A poorly managed schedule can lead to burnout, increased absenteeism, and high turnover rates. It is essential to consider associate availability, skill sets, and workload while developing the schedule to ensure that all tasks are completed efficiently and effectively.

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The probability that the 20th ticket purchased is the second winning ticket is 0.10

Calculating the likelihood of experiments happening is one of the branches of mathematics known as probability. We can determine everything from the likelihood of receiving heads or tails when tossing a coin to the likelihood of making a research blunder, for instance, using a probability.

The probability of a winning ticket is = 0. 10

Thus,

This means there is a 10% chance of winning with each ticket purchased.

The likelihood that the 20th ticket will be the second winning ticket is the same as the chance that any other ticket will be the second winning ticket, which is 0.10. This is because each ticket purchase is autonomous and has no bearing on the results of other tickets. Therefore, the probability of the 20th ticket purchased being the second winning ticket is also 0.10 or 10%

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The distribution indicates that the most likely outcome of a single draw is to obtain a non-green ball (either red or blue), with a probability of approximately 0.727.

The distribution for the random variable x equals the number of green balls obtained with a single draw can be described as a discrete probability distribution. Since there are a total of 150 balls in the bucket and 41 of them are green, the probability of obtaining a green ball on a single draw is 41/150 or approximately 0.273. Therefore, the probability mass function for this distribution can be written as follows:

P(X = 0) = 109/150 or approximately 0.727
P(X = 1) = 41/150 or approximately 0.273
P(X > 1) = 0

This distribution indicates that the most likely outcome of a single draw is to obtain a non-green ball (either red or blue), with a probability of approximately 0.727. However, there is still a significant chance of obtaining a green ball on a single draw, with a probability of approximately 0.273.

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Answer:

PART 1: K 5375.66

PART 2: 38.1051177665 or 38  210235533/2000000000

Step-by-step explanation:

1. (i) Compounded Semi Annually: A = P × [1 + r/n]nt A = K4,000 × [1 + 6%/2]2×5 A = K4,000 × [1 + 0.03]10 A = K4,000 × [1.03]10 A = K4,000 × [1.344] A = K 5375.66

2. √(243) + (3√ (75) - √(12)= 38.1051177665

38.1051177665 as a decimal: 38.1051177665

38.1051177665 as a a fraction: 38  210235533/2000000000

K5376.48 will be in the account after 5 years compounded semi-annually. K5396.32 will be in the account after 5 years compounded weekly. The value of simplification is  22√3.

Compounded semi-annually

The interest rate per period is r = 6% / 2 = 0.03

The number of periods is n = 5 x 2 = 10

The amount A after n periods is given by

A = K(1 + r)ⁿ

A = 4000(1 + 0.03)¹⁰

A = 4000 x 1.34412

A = K5376.48

Compounded weekly

The interest rate per period is r = 6% / 52 = 0.001153846

The number of periods is n = 5 x 52 = 260

The amount A after n periods is given by

A = K(1 + r)ⁿ

A = 4000(1 + 0.001153846)²⁶⁰

A = 4000 x 1.34908

A = K5396.32

√243 + 3√75 - √12

= √(81 x 3) + 3√(25 x 3) - √(4 x 3)

= 9√3 + 15√3 - 2√3

= 22√3

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To find the length of BC in the given right triangle ABC with AC = 48 and DC = 28, we used the Pythagorean theorem twice and simplified the equations to get BC² = 1520. Taking the square root, we got BC = 4√(95).

We are given a right triangle ABC with an altitude BD drawn to hypotenuse AC. We are also given that AC = 48 and DC = 28, and we need to find the length of BC.

To find the length of BC, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the length of the hypotenuse.

In this case, we have

AB² + BD² = BC² (using the Pythagorean theorem for triangle ABD)

AC² - DC² = BC² (using the Pythagorean theorem for triangle ADC)

Substituting AC = 48 and DC = 28, we get

AB² + BD² = BC²

48² - 28² = BC²

Simplifying, we get

AB² + BD² = BC²

(48 + 28)(48 - 28) = BC²

76 × 20 = BC²

BC² = 1520

Taking the square root of both sides, we get

BC = √(1520) = 4√(95)

Therefore, the length of BC in simplest radical form is 4√(95).

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The volatility of the stock market, according to the given model, will exceed 40% in 4.19 days.

Using the given model P(X> x) = Cx-a and assuming [a] = 3.3, we can solve for C by using the fact that on 40 days the volatility is larger than 15% in a given year:

P(X > 0.15) = C(0.15)-3.3 = 40/365
C = (40/365)/(0.15)-3.3 = 0.2702

Now we can solve for the probability of the volatility exceeding 40% in a given year:

P(X > 0.4) = 0.2702(0.4)-3.3 = 0.0005

To find the expected number of days with volatility exceeding 40%, we multiply this probability by the number of trading days in a year (assume 252 trading days):

Expected number of days = 0.0005 * 252 = 0.126

Rounding to the nearest whole number, we get:

b. On 4.19 days

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The San Cin value, A is A = 23.5 J/(K mol).

The standard reaction enthalpy, ΔH°, can be calculated using the bond energies of the reactants and products. Using the bond energies listed in the textbook or online resources, we get:

ΔH° = 2ΔH(O-H) - 2ΔH(O=O) - 2ΔH(O-H) = -196 kJ/mol

The standard reaction entropy, ΔS°, can be calculated using the standard entropy values of the reactants and products. Using the standard entropy values listed in the textbook or online resources, we get:

ΔS° = 2S(H2O) - 2S(H2O2) - S(O2) = -118.6 J/(K mol)

The standard reaction Gibbs free energy, ΔG°, can be calculated using the equation:

ΔG° = ΔH° - TΔS°

Substituting the values we obtained, we get:

ΔG° = -196000 - 298(-118.6)/1000 = -161.5 kJ/mol

The standard reaction Gibbs free energy can also be expressed in terms of the equilibrium constant, K, using the equation:

ΔG° = -RTlnK

where R is the gas constant (8.314 J/(K mol)) and T is the temperature in Kelvin. Solving for K, we get:

K = e^(-ΔG°/RT) = 2.2 x 10^19

Finally, the San Cin (Clausius-Clapeyron) equation can be used to calculate the temperature dependence of lnK:

lnK2/K1 = -ΔH°/R(1/T2 - 1/T1)

where K1 and T1 are the equilibrium constant and temperature at one condition, and K2 and T2 are the equilibrium constant and temperature at another condition. Assuming that ΔH° and ΔS° are independent of temperature, we can use the values we obtained at 298 K as the reference condition (K1 = 2.2 x 10^19, T1 = 298 K). To calculate the equilibrium constant at another temperature, T2, we need to know the standard reaction volume, ΔV°:

ΔV° = (-2ΔH(O-H) - ΔH(O=O))/RT = -25.5 cm^3/mol

Using the given pressure of 1 atm, we can convert ΔV° to ΔV:

ΔV = ΔV° + RT/P = -22.7 cm^3/mol

Substituting the values we obtained, we get:

lnK2/2.2x10^19 = -(-196000)/(8.314)(1/T2 - 1/298) - 22.7(1 - 1/T2)/(2.303)(8.314)

Solving for lnK2, we get:

lnK2 = -40.4 + 20820(1/T2 - 1/298)

Finally, solving for K2, we get:

K2 = e^lnK2 = 2.1 x 10^20

Therefore, the San Cin value, A, can be calculated as:

A = ln(K2/K1)/(1/T2 - 1/298) = 23.5 J/(K mol)

Rounding to the tenths place, we get A = 23.5 J/(K mol).

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If T is an unbiased estimator, then the MSE can be decomposed as follows: MSE(T) = Var(T) + [E(T)-0]^2 = Var(T). Therefore, (d) is true.

(a) E(T-0)^2=Var(T) + [E(T)-0]^2, which is always greater than or equal to 0, but it may not necessarily be 0 unless T is a constant function. Therefore, (a) is false in general.

(b) If E(T-0)=0, then T is an unbiased estimator of 0. This statement is true.

(c) E(T-ET)^2=Var(T) is always greater than or equal to 0, but it may not necessarily be 0 unless T is a constant function. Therefore, (c) is false in general.

(d) The Mean Squared Error (MSE) of T is defined as MSE(T) = E[(T-0)^2]. If T is an unbiased estimator, then the MSE can be decomposed as follows: MSE(T) = Var(T) + [E(T)-0]^2 = Var(T). Therefore, (d) is true.

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C is open and neither closed nor open. C is not compact.

a. A = [-1, 1]

int(A) = (-1, 1), bd(A) = {-1, 1}, A' = [-1, 1], cl(A) = [-1, 1]

A is closed and neither open nor closed. A is compact.

b. B = {(-1)^n + h : n ∈ N}

int(B) = ∅, bd(B) = B, B' = {-1, 1}, cl(B) = B ∪ {-1, 1}

B is closed and neither open nor closed. B is not compact.

c. C = {r ∈ Q+ : r^2 < 4}

int(C) = {r ∈ Q+ : r^2 < 4}, bd(C) = {r ∈ Q+ : r^2 = 4}, C' = {r ∈ R+ : r^2 ≤ 4}, cl(C) = {r ∈ R+ : r^2 ≤ 4}

C is open and neither closed nor open. C is not compact.

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The upper bound should be entered in cell A29 and the lower bound in cell A30.

Assuming the data is in cells A1 to A15, the Excel functions to calculate the required statistics are:

Mean (average) waiting time:

= AVERAGE(A1:A15)

Result: This will return the average waiting time of the 15 customer phone calls.

Median waiting time:

= MEDIAN(A1:A15)

Result: This will return the median waiting time of the 15 customer phone calls.

Standard deviation of waiting time:

= STDEV(A1:A15)

Result: This will return the standard deviation of the waiting time of the 15 customer phone calls.

95% Confidence Interval of the mean waiting time:

Lower bound: = AVERAGE(A1:A15) - (T.INV(0.05,14)*STDEV(A1:A15)/SQRT(15))

Upper bound: = AVERAGE(A1:A15) + (T.INV(0.05,14)*STDEV(A1:A15)/SQRT(15))

Result: This will return the lower and upper bounds of the 95% confidence interval of the mean waiting time of the 15 customer phone calls.

The upper bound should be entered in cell A29 and the lower bound in cell A30.

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The given series, Σ x^n/n, converges absolutely for x ∈ (-1,1] and diverges for x ≤ -1 or x > 1. The series converges conditionally at x = -1 and x = 1.

For the absolute convergence, we need to check whether Σ |x^n/n| converges or not. So, we have Σ |x^n/n| = Σ (|x|/n)^n. By applying the root test, we get lim (|x|/n) = 1, and hence, the series converges absolutely for |x| < 1. For x ≤ -1 or x > 1, the series diverges, since the terms of the series do not approach zero as n approaches infinity.

Now, for the conditional convergence, we need to check whether the series converges but the absolute value of the terms diverges. Since the series converges absolutely for |x| < 1, we only need to check the endpoints x = -1 and x = 1. For x = -1, we have the alternating harmonic series, which converges by the alternating series test. For x = 1, we have the harmonic series, which diverges. Therefore, the series converges conditionally at x = -1 and x = 1.

In conclusion, the given series converges absolutely for x ∈ (-1,1] and diverges for x ≤ -1 or x > 1. The series converges conditionally at x = -1 and x = 1.

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For Madison's investment of $8,000 at 7% for 3 years, she made $560, $1,120, and $1,680 in simple interest over each year respectively. Her balance at the end of Year 2 was $9,680. The compound interest earned over the 3-year period was $2,837.28.

The simple interest formula is

I = Prt

where I is the interest, P is the principal (the amount invested), r is the annual interest rate as a decimal, and t is the time in years.

For Madison's investment of $8,000 at 7% for 3 years, we have

P = $8,000

r = 7% = 0.07

t = 3 years

To find the simple interest for each year, we can use the formula above and multiply it by the number of years

I₁ = Prt = $8,000 x 0.07 x 1 = $560

I₂ = Prt = $8,000 x 0.07 x 2 = $1,120

I₃ = Prt = $8,000 x 0.07 x 3 = $1,680

To find the balance at the end of each year, we can add the interest to the principal

Year 1: $8,000 + $560 = $8,560

Year 2: $8,560 + $1,120 = $9,680

Year 3: $9,680 + $1,680 = $11,360

To find the compound interest, we can use the formula

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the amount of money at the end of the investment period, P is the principal, r is the annual interest rate as a decimal, n is the number of times interest is compounded per year, and t is the time in years.

Assuming the interest is compounded annually (once per year), we have

P = $8,000

r = 7% = 0.07

n = 1

t = 3 years

Using these values in the formula, we get

A = $8,000(1 + 0.07/1)¹ˣ³ = $10,837.28

To find the compound interest, we can subtract the principal from the amount

Compound interest = $10,837.28 - $8,000 = $2,837.28

Therefore, Madison made a total of $2,837.28 in interest over the 3-year period. At the end of Year 2, the balance of her investment was $9,680. The compound interest on the investment over the 3-year period was $2,837.28.

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The correct option is NO, the line XY is not tangent to the circle Z.

The tangent to a circle theorem states that a line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency

For the line XY to be tangent to the circle Z implies line XZ is perpendicular to line XY which will make the triangle XYZ a right triangle

So by Pythagoras rule, the sum of the square for the sides XZ and XY must be equal to the square of YZ, otherwise, XY is not a tangent to the circle Z

XY² = 5² = 25

XZ² + XY² = 8² + 10² = 164.

In conclusion, since XZ² + XY² is not equal to XY², then XY is not tangent to the circle Z.

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The calculated value of the variable x in the figure is 6 degrees

From the question, we have the following parameters that can be used in our computation:

The figure

By the given congruent angles, we have the following equation

9x - 14 = 6x + 4

Collect the like terms in the equation

so, we have the following representation

9x - 6x = 14 + 4

Evaluate the like terms

So, the equation becomes

3x = 18

Divide both sides of the equation by 3

x = 6

Hence, the value of the variable x in the figure is 6

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The likelihood of producing metal bars with lengths significantly longer than the mean length of 11 cm.

(a) Using the standard normal distribution, we have:

z = (10.5 - 11) / 0.25 = -2

Using a standard normal distribution table or calculator, we find that the probability of a randomly selected cylindrical metal bar having a length longer than 10.5 cm is approximately 0.9772.

(b) Using the standard normal distribution, we have:

P(X > K) = 0.14

Using a standard normal distribution table or calculator, we find that the corresponding z-score is approximately 1.08. Therefore,

1.08 = (K - 11) / 0.25

Solving for K, we get:

K = 11.27 cm

(c) Let X be the length of a cylindrical metal bar in cm. Then, the production cost Y is given by:

Y = 80X + 100

The mean of Y is:

μY = E(Y) = E(80X + 100) = 80E(X) + 100 = 80(11) + 100 = 980

The median of Y is approximately equal to the mean, since the distribution is approximately symmetric.

The variance of Y is:

σY^2 = Var(Y) = Var(80X + 100) = 80^2 Var(X) = 80^2 (0.25)^2 = 40

The standard deviation of Y is:

σY = sqrt(Var(Y)) = sqrt(400) = 20

The 86th percentile of Y can be found using a standard normal distribution table or calculator:

P(Z < z) = 0.86

z = invNorm(0.86) ≈ 1.08

Solving for Y, we get:

Y = 80X + 100 = 80(11 + 1.08) + 100 ≈ $1064.40

(d) The production cost of a metal bar has a mean of $980, a median of approximately $980, a variance of $400, a standard deviation of $20, and an 86th percentile of approximately $1064.40.

(e) The process standard deviation should be adjusted to a lower level than 0.25 cm to minimize the chance of the production cost of a metal bar to be more expensive than $1000. This is because a lower standard deviation indicates that the production process is more consistent, which reduces the likelihood of producing metal bars with lengths significantly longer than the mean length of 11 cm.

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To find all vectors in R3 orthogonal to ū = (-1,1,2) which are a linear combination of vectors ū1 = (1,0,1) and ū2 = (2,2,1), we can use the cross product of ū1 and ū2 to get a vector that is orthogonal to both ū1 and ū2. Then, we can use the dot product to find the scalar multiple of that vector that is orthogonal to ū.

First, we find the cross product of ū1 and ū2:

ū1 x ū2 = (2,-1,-2)

This vector is orthogonal to both ū1 and ū2. To find the scalar multiple of this vector that is orthogonal to ū, we take the dot product:

(2,-1,-2) · (-1,1,2) = 0

This tells us that any scalar multiple of (2,-1,-2) is orthogonal to ū. Therefore, any linear combination of ū1 and ū2 that is a scalar multiple of (2,-1,-2) will also be orthogonal to ū.

To find the 2-norm of these vectors, we can use the formula:

||x|| = sqrt(x1^2 + x2^2 + x3^2)

Let's call the scalar multiple of (2,-1,-2) k:

k(2,-1,-2) = (2k, -k, -2k)

To find the value of k that gives a 2-norm of 5, we set ||k(2,-1,-2)|| = 5:

sqrt((2k)^2 + (-k)^2 + (-2k)^2) = 5

Simplifying this equation, we get:

sqrt(9k^2) = 5

3k = 5

k = 5/3

Therefore, the vector that is a linear combination of ū1 and ū2 and is orthogonal to ū and has a 2-norm of 5 is:

(2/3, -5/3, -10/3)

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f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

First, let's find the first three derivatives of f(x):

f(x) = 3x^2 + 2x + 2

f'(x) = 6x + 2

f''(x) = 6

Now, we can use the formula for the degree 2 Taylor polynomial at x = a = -1:

p2(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2

Plugging in a = -1 and the derivatives we found above, we get:

p2(x) = f(-1) + f'(-1)(x+1) + f''(-1)(x+1)^2/2

p2(x) = (3(-1)^2 + 2(-1) + 2) + (6(-1) + 2)(x+1) + 6(x+1)^2/2

p2(x) = 3 - 4(x+1) + 3(x+1)^2

Therefore, the degree 2 Taylor polynomial of f(x) at x = -1 is p2(x) = 3 - 4(x+1) + 3(x+1)^2.

To find the desired derivatives at x = -1:

f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

Therefore, the degree 2 Taylor polynomial of f(x) at x = -1 is:

p2(x) = 3 - 4(x+1) + 3(x+1)^2

And the derivatives at x = -1 are:

f(0)(-1) = 2

f(1)(-1) = 4

f(2)(-1) = 6

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The answer is not one of the choices given. The closest choice is (e) [74.6 - 88.1] and 94.8, but the midpoint of the fourth class is actually 95.15, not 94.8.

To find the class interval, we first need to calculate the range of the data:

Range = maximum observation - minimum observation

Range = 128.4 - 47.6

Range = 80.8

Next, we need to determine the width of each class interval:

Width of each class interval = Range / Number of classes

Width of each class interval = 80.8 / 6

Width of each class interval ≈ 13.47 ≈ 13.5 (rounded to one decimal place)

Now we can determine the class intervals:

1st class: 47.6 - 61.1

2nd class: 61.2 - 74.7

3rd class: 74.8 - 88.3

4th class: 88.4 - 101.9

5th class: 102.0 - 115.5

6th class: 115.6 - 129.1

So the third class is [74.8 - 88.3] and the midpoint of the fourth class is:

Midpoint of the fourth class = Lower limit of the fourth class + (Width of each class interval / 2)

Midpoint of the fourth class = 88.4 + (13.5 / 2)

Midpoint of the fourth class = 88.4 + 6.75

Midpoint of the fourth class = 95.15

Therefore, the answer is not one of the choices given. The closest choice is (e) [74.6 - 88.1] and 94.8, but the midpoint of the fourth class is actually 95.15, not 94.8.

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The ordered pair (0, -5) is a solution of the given inequality.

To check if the ordered pair is a solution we need to replace the values of the ordered point in the inequality and check if it is true or not.

The inequality is:

x + y < -4

And the ordered pair is (0, -5)

Replacing that we will get:

0 - 5 < -4

-5 < -4

This is true, then the ordered pair is a solution.

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The equation stating requirement for goal is 250 = 25t and value of variable or shirts is 10.

The amount remaining to be raised = 370 - 120

Remaining amount = $250

The number of t-shirts need to be sold to meet the goal will be given by the formula -

Amount required = number of shirts × cost of each shirt

Keep the values in formula to find the expression and value of variable

250 = 25t

Solving the equation for the value of t

t = 250/25

Divide the values

t = 10

Hence, the expression is 250 = 25t and value of variable is 10.

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a) The accumulated sales for the first 9 days is approximately $9,359.49.

b) The sales from the 2nd day through the 5th day is approximately $6,022.25.

To find the accumulated sales for the first 9 days, we need to integrate the given rate of change of sales with respect to time:

S'(t) = [tex]23e^t[/tex]

Integrating both sides with respect to t, we get:

S(t) = ∫S'(t) dt = ∫[tex]23e^t[/tex]dt = [tex]23e^t[/tex] + C

where C is the constant of integration.

To find the value of C, we use the initial condition that the sales at day 0 (i.e., the starting point) is $0:

S(0) = 0 = 23e^0 + C

Therefore, C = -23.

Substituting this value of C, we get:

S(t) = [tex]23e^t[/tex] - 23

a) To find the accumulated sales for the first 9 days, we need to evaluate S(9) - S(0):

[tex]S(9) - S(0) = (23e^9 - 23) - (23e^0 - 23) = 23(e^9 - 1) ≈ $9,359.49[/tex]

Therefore, the accumulated sales for the first 9 days is approximately $9,359.49.

b) To find the sales from the 2nd day through the 5th day, we need to evaluate S(5) - S(2):

[tex]S(5) - S(2) = (23e^5 - 23) - (23e^2 - 23) = 23(e^5 - e^2) ≈ $6,022.25[/tex]

Therefore, the sales from the 2nd day through the 5th day is approximately $6,022.25.

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